Why is it called Cramer's rule?
In 1750, Gabriel Cramer published a paper outlining the famous technique that today bears his name: Cramer's rule. The Swiss genius had realized that determinants can be used to solve systems of linear equations.
At an age of merely 18, Cramer received his doctorate degree at the University of Geneva. The institution was so impressed by the young mathematician's abilities, that they created a new position for him as co-chairman of mathematics at the university.
As it turned out, this was a clever move by the University and benefited the whole city of Geneva.
Cramer stayed with the University for the remainder of his life, where he reformed the education so that mathematics would be taught in French and not only in Latin, thereby reaching a wider audience.
What is the definition of the determinant?
The determinant of a matrix is a scalar value denoted by or . For to exist, the matrix must be square, and if it is, reveals information about the solutions to the system of equations the matrix constitutes.
If the determinant is zero, we have infinitely many or no solutions to the given system. All other values mean that there is a unique solution.
For matrices whose determinants are zero, we can be sure that there are either an infinite number of solutions or none of them. Moreover, a non-zero determinant will always give rise to a unique solution.
The value of the determinant is also closely related to the inverse of a matrix. If and only if a matrix has a non-zero determinant, it is invertible and we can use the determinant to find the inverse matrix.
Furthermore, the determinant gives the scaling factor of the linear transformation that a matrix describes.
How to find the determinant of a square matrix?
To find the determinant of a square matrix (it must be square), we can use methods like the Leibniz formula or Laplace expansion, which will always work. There are however short-cuts we can use in certain cases.
If is a matrix, its determinant can be found quickly using the following formula:
In case we instead are dealing with a matrix, this is the formula to use:
More about the determinant
The determinant is a scalar and is noted:
The determinant can be introduced both late and early in a course in linear algebra. As for what it is, students are traditionally first introduced to how the determinant is calculated and later to the practical connection and its geometric interpretation.
We choose to do the opposite.
A practical connection
The determinant tells whether a linear system of equations has solutions or not. Remember the three cases; unique solution, infinitely many solutions or no solutions.
If the determinant is zero, the system has "infinitely many solutions" or "no solutions".
If the determinant is non-zero, the system has a unique solution.
A geometric interpretation
The determinant is interpreted geometrically as the scale factor for a linear transformation, to which the beginner unfortunately has not usually been introduced when determinant calculations are necessary.
In short, each matrix multiplication is a linear transformation, but from a practical perspective, it can be said that a linear transformation is a matrix that multiplies with a vector to obtain a desired result.
A simple example would be a linear transformation that rotates clockwise by the angle and doubles its length. Then the scale factor, that is, the determinant, of would be .
The definition of the determinant of a -matrix forms the basis for calculating the determinant of an -matrix.
whereupon the definition of the determinant is:
The algorithm for calculating the determinant of a -matrix is made using the sum of three -determinants. We produce these by expanding a single row, or column, in the determinant (called cofactor expansion).
and then it applies that the determinant of is:
where we have made a row expansion of the first row, because the scalars of each -matrix are just the elements from the first row.
Now we go through how the expansion is done. Consider the determinant of :
We start by expanding along the first line and start with the first element :
The expansion then takes place by selecting the row and column of the current element to extract the remaining elements as a -determinant multiplied by :
We move on to the next element along the first line, , and get:
Note that the expansion around comes with a minus sign! We'll return to that shortly.
Now we continue with the next, and last, element to expand: .
Note that the element comes with a plus sign!
We now end the calculation using the definition of the -determinant:
Which concludes the formula for the -determinant, as well as the algorithm that makes the definition easy to remember instead of learning the formula by heart (something required to advance from beginner-status).
An alternative formula
The method above can easily be extended analogously to larger matrices, which is why we started with it. However, there is an alternative algorithm that applies to the -determinant, which visually resembles the definition of the -determinant:
If we extend this mindset, we get a method that works, but only works for calculating -determinants. The method is called Sarru's rule.
The calculation of the determinant, regardless of the dimensions of matrix , is done analogously as for the -determinant - we can express it as an algorithm for each -determinant. But before we do that, we explain why the element in the calculation of the -determinant had a minus sign.
Consider the -matrix . In that case, each extracted element in its determinant carries with it a plus sign or a minus sign depending on its position, according to the following "chess pattern":
This means that for the -determinant, the hidden sign for each element follows:
For example, if we were to choose to expand along the second column, the product sum would be:
Note that plus and minus signs written within the above determinants should not be made in any calculation, but have now been made for educational purposes only.
The general form of an expansion along a line (cofactor expansion) for the determinant of an -matrix can be written as:
where is each element in selected row , and is a cofactor, which is the -determinant of the other elements that do not divide a row or column with the corresponding .
Algorithm for nxn determinant
Select a row, or column, to be expanded into the product sum of matrix elements and -determinants
For each element in the selected row / column:
Extract an element with the plus sign or minus sign it carries and multiply by the -determinant of the elements that do not share a row or column with the extracted element
Repeat until all elements in the selected row / column are extracted
\item Repeat the above steps until the last product sum contains only -determinants.
The algorithm shows that the calculation of a determinant can be extremely laborious if the dimension is high.
Note however, the advantage of extracting a row, or column, whose many elements are zero! This means that the developed product sum is greatly reduced. For example, as in:
If the determinant you calculate lacks 0-elements, or does not have enough to greatly simplify the calculation, you can, like Gauss-Jordan, row reduce the determinant matrix without changing the determinant. This, and more features, are discussed in the next section.
Adjoint of matrix
The adjoint of matrix is based on the cofactor expansions of . This becomes interesting in a theorem for the expression of , if the inverse exists. Our definition of the adjoint of matrix is:
If is an -matrix and is the cofactor of , then it follows that the matrix:
is called the cofactor matrix of A. The transpose of this matrix is called the adjoint of matrix of and is noted as .
Using the adjoint of matrix of , we can very easily express if the inverse exists using the following theorem, which we leave unproven.
If is an invertible matrix, then:
We will now show examples of all this. Let be the following invertible matrix:
of which the cofactors become:
and thus the cofactor matrix and the adjoint of matrix become the following:
We start with a useful theorem on the laws to row reduce the determinant matrix before a row expansion to maximize the number of 0-elements.
For each -matrix it applies that:
If matrix is the result of a scalar multiplied by a row, or column, in matrix , then:
If matrix is the result of two rows, or columns, having changed places in , it applies that:
If matrix is the result of a multiple of a row or column in matrix being added to another row or column, then:
The proof of the first and third point is a good exercise for the beginner, and a direct proof of a -determinant and a -determinant is enough to make one convinced. To create a sustainable mathematical proof, a proof by induction is recommended.
The second point follows from the determinant's definition with the "chess-patterned" character scheme in the previous section.
With the help of the previous statement, we can get the following statement:
Let be an -matrix.
If two rows or columns are equal, then:
If a row or column can be row reduced to 0, then:
If is a scalar, then:
Now we are ready for the most memorable theorem for students, which is based on the last theorems and the evidence we have presented for how an invertible matrix can be row-reduced to :
A square matrix is invertible if, and only if, .
Assume that can be row-reduced to , then:
Assume the opposite, that can not be row-reduced to , but to . This means that is not invertible because at least two lines in are linearly dependent, and we get at least one zero line in . A single zero row results in:
Another useful theorem for arithmetic is:
If and are square matrices with the same dimensions, then:
The following statement applies to the inverse:
If the matrix is invertible, it applies that:
Remember that . Then we have:
Because , we have:
We end this section by linking a theorem that is introduced with inverse and linear systems of equations along with our insights with the determinant.
Let be an -matrix. Then the following statements apply:
The reduced row echelon form for is
can be expressed as a product of elementary matrices
has only the trivial solution
is consistent for each vector in
has exactly one solution for each vector in
The column vectors of are linearly independent
The row vectors in are linearly independent
For each equation:
where the matrix is invertible, there is a unique solution for each and . Cramer's rule is a statement that, above all, makes it easier to express the solution , since we do not have numbers in the matrix .
Cramer's rule If is a linear system of equations with equations and variables, then the system has a unique solution if, and only if, , whereupon the solution can be expressed as:
where is the matrix where column in is replaced with .
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